respectively.
In Eqs. (4) and (5), \(e = |e|\)  is the carrier charge, \(b= [3\Gamma (1/4)\Gamma (1/2)/2\Gamma (3/4)]\simeq 0.0716\)  [1]}, \(\kappa _{eff}\) 
is the effective dielectric constant accounting for the dielectric constant, \(\kappa _S\)  and \(\kappa _T\) , of the substrate and top/passivation layer at the frequency in the THz range and their thickness, \(\Lambda = (L/l)\tan ^{-1}\frac{1}{\displaystyle \sqrt{(L/l)^2-1}}+\ln [(L/l)+ \sqrt{(L/l)^2-1}] \simeq \ln (2L/l)\)  [2]} (see also [3]}, [4]}, [5]}, [6]}) is a logarithmic factor depending on the device geometrical parameters (capacitance factor),
and
\({\mathcal {J}}_0(s)\)  is the Bessel functions. Generally, \(\kappa _{eff} = (\kappa _S + \kappa _T)/2\) . In the case of a rather thin top/passivation layer with the thickness \(w \ll 2l\) , \(\kappa _{eff} \simeq (\kappa _S + 1)/2)\) .
The factor of 1/2 is because the streamlines of electric field emerge out of the top layer.
